Normalized defining polynomial
\( x^{8} - 4x^{7} + 20x^{6} - 44x^{5} + 98x^{4} - 128x^{3} + 144x^{2} - 96x + 34 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(378535936\) \(\medspace = 2^{20}\cdot 19^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(11.81\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{5/2}19^{1/2}\approx 24.657656011875904$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{85}a^{6}+\frac{7}{17}a^{5}+\frac{2}{85}a^{4}-\frac{6}{85}a^{3}-\frac{12}{85}a^{2}-\frac{33}{85}a-\frac{1}{5}$, $\frac{1}{85}a^{7}-\frac{33}{85}a^{5}+\frac{9}{85}a^{4}+\frac{28}{85}a^{3}-\frac{38}{85}a^{2}+\frac{33}{85}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{2}{85} a^{7} - \frac{19}{85} a^{5} - \frac{18}{85} a^{4} - \frac{56}{85} a^{3} - \frac{9}{85} a^{2} - \frac{66}{85} a + 1 \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{12}{85}a^{7}-\frac{6}{17}a^{6}+\frac{169}{85}a^{5}-\frac{207}{85}a^{4}+\frac{516}{85}a^{3}-\frac{351}{85}a^{2}+\frac{366}{85}a-3$, $\frac{6}{85}a^{7}-\frac{9}{85}a^{6}+\frac{82}{85}a^{5}-\frac{49}{85}a^{4}+\frac{307}{85}a^{3}-\frac{7}{17}a^{2}+\frac{65}{17}a+\frac{9}{5}$, $\frac{9}{85}a^{7}-\frac{22}{85}a^{6}+\frac{123}{85}a^{5}-\frac{133}{85}a^{4}+\frac{299}{85}a^{3}-\frac{78}{85}a^{2}+\frac{3}{85}a+\frac{7}{5}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 55.0761610673 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 55.0761610673 \cdot 1}{8\cdot\sqrt{378535936}}\cr\approx \mathstrut & 0.551492474014 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 10 conjugacy class representatives for $Q_8:C_2$ |
Character table for $Q_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 16 |
Degree 8 siblings: | 8.4.136651472896.2, 8.0.8540717056.3 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.20.55 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 2$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
\(19\) | 19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.76.2t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | \(\Q(\sqrt{19}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.152.2t1.b.a | $1$ | $ 2^{3} \cdot 19 $ | \(\Q(\sqrt{-38}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.152.2t1.a.a | $1$ | $ 2^{3} \cdot 19 $ | \(\Q(\sqrt{38}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.1216.8t11.a.a | $2$ | $ 2^{6} \cdot 19 $ | 8.0.378535936.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
* | 2.1216.8t11.a.b | $2$ | $ 2^{6} \cdot 19 $ | 8.0.378535936.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |